In Exercises 5−8, r(t) is the position of a particle in the xy-plane at time t. Find an equation in x and y whose graph is the path of the particle. Then find the particle’s velocity and acceleration
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University Calculus: Early Transcendentals (4th Edition)
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- Find the acceleration of the particle at t = 0 with the position function r(t)=e6ti+e8tjarrow_forwardSuppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 4x2 − 3xy + xyz (a) Find the rate of change of the potential at P(2, 4, 5) in the direction of the vector v = i + j − k. (b) In which direction does V change most rapidly at P? (c) What is the maximum rate of change at P? Thank youarrow_forwardAn object travels along the curve C with the position function r(t)= t^2i + 2tj + tk for t ≥ 0 (a) Compute the velocity vector at time t=2. (b) Compute the acceleration vector at time t=2. (c) Compute the equation of the line that is tangent to the curve when t=2. Write the answer in the symmetric form.arrow_forward
- Rather than the linear relationship of Eq. (1.7), you might choose to model the upward force on the parachutist as a second-order relationship. FU = −c'v2, where c' = a second-order drag coefficient (kg/m). a. Using calculus, obtain the closed-form solution for the casewhere the jumper is initially at rest (v = 0 at t = 0).b. Repeat the numerical calculation in Example 1.2 with thesame initial condition and parameter values. Use a value of0.225 kg/m for c' .arrow_forwardSketch the path r(t) =〈t^2, t^3〉together with the velocity and acceleration vectors at t = 1.arrow_forwardA particle moves with position function r(t)= (t 2, t 2, t 3 ) Find the tangential and normal components of acceleration.arrow_forward
- Find all solutions to r'(t) = v with initial condition r(1) = w, where v and w are constant vectors in R^3.arrow_forwardSuppose an object moves along the x-axis so that its position at time t is x = −t +t3/6 (a) Find the velocity, v(t) = x˙ (t), of the object. (b) What is v(0)? What does this say about the direction of motion of the object at time t = 0? (c) When is the object at the origin? What is the velocity of the object when it is at the origin?arrow_forwardGiven the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, find the body's position at time t.a = 9.8, initial velocity = 5, initial position = -1 r = 4.9t2 + 5t - 1 r = -4.9t2 - 5t - 1 r = 9.8t2 + 5t - 1 r = 4.9t2 + 5tarrow_forward
- ind the orthogonal trajectories of the family of curve x2/3 +y2/3 =a2/3 where "a" is the parameterarrow_forwardFind the solution for the damped motion. my'' + cy' + ky = 0 with m = 10, k = 90, y(0) = 0.16, y'(0) = 0, c = 10arrow_forwardGiven the acceleration and initial velocity of a body moving along a coordinate line at time t, find the body's position at time t.acceleration = -8t + 5, initial velocity = 2 v = -8t2 + 5t + 2 v = t3 + t2 + 2t + C v = -4t2 + 5t + 2 v = -4t2 + 5t + 1arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage